Goodness of fit and variable selection in the fuzzy multiple linear regression

In performing a fuzzy multiple linear regression model, important topics are: to measure the fitting quality of the model and to find the “best” set of input variables that explain the variation in the observed system responses. In this paper, by considering an exploratory approach, to express the quality of fit of a fuzzy linear regression model, a coefficient of multiple determination R2 for symmetrical fuzzy variable has been suggested. Furthermore, for overcoming the inconveniences of R2 an adjusted version of R2 (denoted by ) has been defined. For measuring the fitting performances of the estimated model, a fuzzy extension of another goodness of fit measure, the so-called Mallows index (Cp), has been considered. All the proposed fitting measures have been utilized for selecting suitably the input variables of a fuzzy linear regression model. To this purpose, some variable selection procedures based on R2, and Cp have been suitably extended in a fuzzy framework. To explain the efficacy of the goodness of fit measures and the variable selection criteria some examples are also shown.